A string can only provide force in its own direction as tension. So the tension force would be $T \cos(37) \hat x + T \cos (37) \hat y$ where $T$ is the total tension force, for now is unknown.
The vertical force is obviously $-mg \hat y$.
Now you need to make zero net force IN THE DIRECTION of the string (sentence corrected), because the object does not move (accelerate) in that direction so net force is zero. Think about those two vectors and try to come up with an equation, and write in comments. I’m not allowed to do it for you.
So in this case the magnitude of the component of gravity that acts along the direction of the string is equal to T. in some sense it creates the counterforce of tension.
If the board moves up we can set-up an inertial frame. Read this answer first: Non-inertial reference frame, a pendulum in an accelerating car
What we will do is make a reference frame from the point of view of the board, where we move with the board. Try something on that in the comments or in a new question.